1 Grassmannian Codes with New Distance Measures for Network Coding TUVI ETZION, Fellow, IEEE HUI ZHANG Abstract—Grassmannian codes are known to be useful in network has N receivers, each one demands all the h error-correction for random network coding. Recently, they messages of the source to be transmitted in one round of were used to prove that vector network codes outperform a network use. An up-to-date survey on network coding scalar linear network codes, on multicast networks, with respect to the alphabet size. The multicast networks which for multicast networks can be found for example in [21]. were used for this purpose are generalized combination Kotter¨ and Medard´ [29] provided an algebraic formulation networks. In both the scalar and the vector network coding for the network coding problem: for a given network, find solutions, the subspace distance is used as the distance coding coefficients for each edge, whose starting vertex measure for the codes which solve the network coding has in-degree greater than one. These coding coefficients problem in the generalized combination networks. In this work we show that the subspace distance can be replaced are multiplied by the symbols received at the starting node with two other possible distance measures which generalize of the edge and these products are added together. These the subspace distance. These two distance measures are coefficients should be chosen in a way that each receiver shown to be equivalent under an orthogonal transformation. can recover the h messages from its received symbols on It is proved that the Grassmannian codes with the new its incoming edges. This sequence of coding coefficients distance measures generalize the Grassmannian codes with the subspace distance and the subspace designs with the at each such edge is called the local coding vector and strength of the design. Furthermore, optimal Grassmannian the edge is called a coding point. Such an assignment codes with the new distance measures have minimal re- of coding coefficients for all such edges in the network quirements for network coding solutions of some generalized is called a solution for the network and the network is combination networks. The coding problems related to these called solvable. It is easy to verify that the information two distance measures, especially with respect to network coding, are discussed. Finally, by using these new concepts on each edge is a linear combination of the h messages. it is proved that codes in the Hamming scheme form a The vector of length h of these coefficients of this linear subfamily of the Grassmannian codes. combination is called the global coding vector. From the global coding vectors and the symbols on its incoming edges, the receiver should recover the h messages, by Index Terms—Distance measures, generalized combination networks, Grassmannian codes, network coding. solving a set of h linearly independent equations. The coding coefficients defined in this way are scalars and the solution is a scalar linear solution. Ebrahimi and Fragouli [7] have extended this algebraic approach to I. INTRODUCTION vector network coding. In the setting of vector network ETWORK coding has been attracting increasing coding, the messages of the source are vectors of length ` N attention in the last fifteen years. The seminal work over Fq and the coding coefficients are ` × ` matrices of Ahlswede, Cai, Li, and Yeung [1] and Li, Yeung, over Fq. A set of matrices, which have the role of the and Cai [24] introduced the basic concepts of network coefficients of these vector messages, such that all the arXiv:1801.02329v7 [cs.IT] 8 Feb 2019 coding and how network coding outperforms the well- receivers can recover their requested information, is called known routing. The class of networks which are mainly a vector solution. Also in the setting of vector network studied is the class of multicast networks and these are coding we distinguish between the local coding vectors also the target of this work. A multicast network is a and the global coding vectors. There is a third type of directed acyclic graph with one source. The source has network coding solution, a scalar nonlinear network code. Again, in each coding point there is a function of the h messages, which are scalars over a finite field Fq. The symbols received at the starting node of the coding point. Department of Computer Science, Technion, Haifa 3200003, Israel, This function can be linear or nonlinear. There is clearly a e-mail: [email protected]. hierarchy, where a scalar linear solution can be translated Nanyang Technological University, Singapore, e-mail: [email protected]. Part of the research was performed to a vector solution, and a vector solution can be translated while the author was with the Department of Computer Science, to a scalar nonlinear solution. Technion, Haifa 3200003, supported in part by a fellowship of the Israel Council of Higher Education. The alphabet size of the solution is an important Parts of this work have been presented at the IEEE International Symposium on Information Theory 2018, Vail, Colorado, U.S.A., June parameter that directly influences the complexity of the 2018. calculations at the network nodes and as a consequence the 2 performance of the network. A comparison between the measures on Grassmannian codes which generalize the required alphabet size for a scalar linear solution, a vector Grassmannian distance. We discuss the maximum sizes solution, and a scalar nonlinear solution, of the same of Grassmannian codes with the new distance measures multicast network is an important problem. It was proved and analyse these bounds from a few different point of in [18], [19] that there are multicast networks on which view. We explore the connection between these codes a vector network coding solution with vectors of length and related generalized combination networks. Our ex- ` over Fq outperforms any scalar linear network coding position will derive some interesting properties of these solution, i.e. the scalar solution requires an alphabet of codes with respect to the traditional Grassmannian codes ` size qs, where qs > q . The proof used a family of and some subspace designs. We will show, using a few networks called the generalized combination networks, different approaches, that codes in the Hamming space where the combination networks were defined and used form a subfamily of the Grassmannian codes. Some other in [34]. interesting connections to subspace designs and codes in Kotter¨ and Kschischang [30] introduced a framework the Hamming scheme will be also explored. for error-correction in random network coding. They have The Grassmannian codes (constant dimension codes) shown that for this purpose the codewords (messages) are are the q-analog of the constant weight codes, where taken as subspaces over a finite field Fq. For this purpose q-analogs replace concepts of subsets by concepts of sub- they have defined the subspace distance. This approach spaces when problems on sets are transferred to problems was mainly applied on subspaces of the same dimension. on subspaces over the finite field Fq. For example, the size For given positive integers n and k, 0 ≤ k ≤ n, the n of a set is replaced by the dimension of a subspace, the Grassmannian Gq(n; k) is the set of all subspaces of Fq binomial coefficients are replaced by the Gaussian coef- whose dimension is k. It is well known that ficients, etc. The Grassmann space is the q-analog of the n n−1 n−k+1 n def (q − 1)(q − 1) ··· (q − 1) Johnson space and the subspace distance is the q-analog = jGq(n; k)j = k k−1 k q (q − 1)(q − 1) ··· (q − 1) of the Hamming distance. The new distance measures are q-analogs of related distances in the Johnson space. n where k q is the q-binomial coefficient (known also as The Johnson scheme J(n; w) consists of all w-subsets the q−ary Gaussian coefficient [45, pp. 325-332]. A code of an n-set (equivalent to binary words on length n and 2 G (n; k) is called a Grassmannian code or a constant C q weight w). The Johnson distance dJ (x; y) between two dimension code. For two subspaces X; Y 2 Gq(n; k) w-subsets x and y is half of the related Hamming distance, the subspace distance is reduced to the Grassmannian i.e., dJ (x; y) , jx n yj. distance defined by The rest of this paper is organized as follows. In def Section II we present the combination network and its gen- dG(X; Y ) = k − dim(X \ Y ) : eralization which was defined in [18], [19]. We discuss the Most of the research on Grassmannian codes motivated family of codes which provide network coding solutions by [30] was in two directions – finding the largest codes for these networks. We will make a brief comparison be- with prescribed minimum Grassmannian distance and tween the related scalar coding solutions and vector cod- looking for designs based on subspaces. To this end, the ing solutions. In Section III we further consider this family quantity Aq(n; 2d; k) was defined as the maximum size of of codes, define two dual distance measures on these a code in Gq(n; k) with minimum Grassmannian distance codes, and show how these codes and the new distance d. There has been extensive work on Grassmannian codes measures defined on them generalize the conventional in the last ten years, e.g. [13], [14], [15], [16], [36] and Grassmannian codes with the Grassmannian distance. We references therein. A related concept is a subspace design show the connection of these codes to subspace designs.